The maximum principle conjecture for the Riesz transform

Let $\mu$ be a nonnegative finite Borel measure with compact support and continuous density with respect to the Lebesgue measure in $\mathbb{R}^d$, and let $0<s<d$. The $s$-Riesz transform (potential) of $\mu$ is

$$R^s\mu(x)=\int\frac{y-x}{|y-x|^{s+1}}\,d\mu(y).$$

Maximum principle conjecture. There is a constant $C=C(d,s)$ such that

$$\sup_{x\in\mathbb{R}^d}|R^s\mu(x)|\le C\sup_{x\in\operatorname{supp}\mu}|R^s\mu(x)|.$$

This relation was known for $d-1\le s<d$ and is proved in the paper for $0<s<1$ and for radial measures, while it remains open in the general case. The conjecture is stated for nonnegative measures; it is false for non-positive measures.